Quantum tomography using state-preparation unitaries

• URL: http://arxiv.org/abs/2207.08800v1
• Date: Mon, 18 Jul 2022 17:56:18 GMT
• Title: Quantum tomography using state-preparation unitaries
• Authors: Joran van Apeldoorn, Arjan Cornelissen, Andr\'as Gily\'en, Giacomo Nannicini
• Abstract summary: We describe algorithms to obtain an approximate classical description of a $d$-dimensional quantum state when given access to a unitary. We show that it takes $widetildeTheta(d/varepsilon)$ applications of the unitary to obtain an $varepsilon$-$ell$-approximation of the state. We give an efficient algorithm for obtaining Schatten $q$-optimal estimates of a rank-$r$ mixed state.
• Score: 0.22940141855172028
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• Abstract: We describe algorithms to obtain an approximate classical description of a $d$-dimensional quantum state when given access to a unitary (and its inverse) that prepares it. For pure states we characterize the query complexity for $\ell_q$-norm error up to logarithmic factors. As a special case, we show that it takes $\widetilde{\Theta}(d/\varepsilon)$ applications of the unitaries to obtain an $\varepsilon$-$\ell_2$-approximation of the state. For mixed states we consider a similar model, where the unitary prepares a purification of the state. In this model we give an efficient algorithm for obtaining Schatten $q$-norm estimates of a rank-$r$ mixed state, giving query upper bounds that are close to optimal. In particular, we show that a trace-norm ($q=1$) estimate can be obtained with $\widetilde{\mathcal{O}}(dr/\varepsilon)$ queries. This improves (assuming our stronger input model) the $\varepsilon$-dependence over the algorithm of Haah et al.\ (2017) that uses a joint measurement on $\widetilde{\mathcal{O}}(dr/\varepsilon^2)$ copies of the state. To our knowledge, the most sample-efficient results for pure-state tomography come from setting the rank to $1$ in generic mixed-state tomography algorithms, which can be computationally demanding. We describe sample-optimal algorithms for pure states that are easy and fast to implement. Along the way we show that an $\ell_\infty$-norm estimate of a normalized vector induces a (slightly worse) $\ell_q$-norm estimate for that vector, without losing a dimension-dependent factor in the precision. We also develop an unbiased and symmetric version of phase estimation, where the probability distribution of the estimate is centered around the true value. Finally, we give an efficient method for estimating multiple expectation values, improving over the recent result by Huggins et al.\ (2021) when the measurement operators do not fully overlap.

Related papers

• Sample-Optimal Quantum Estimators for Pure-State Trace Distance and Fidelity via Samplizer [7.319050391449301]
  Trace distance and infidelity, as basic measures of the closeness of quantum states, are commonly used in quantum state discrimination, certification, and tomography. We present a quantum algorithm that estimates the trace distance and square root fidelity between pure states to within additive error $varepsilon$, given sample access to their identical copies.
  arXiv  Detail & Related papers  (2024-10-28T16:48:21Z)
• Optimal Sketching for Residual Error Estimation for Matrix and Vector Norms [50.15964512954274]
  We study the problem of residual error estimation for matrix and vector norms using a linear sketch. We demonstrate that this gives a substantial advantage empirically, for roughly the same sketch size and accuracy as in previous work. We also show an $Omega(k2/pn1-2/p)$ lower bound for the sparse recovery problem, which is tight up to a $mathrmpoly(log n)$ factor.
  arXiv  Detail & Related papers  (2024-08-16T02:33:07Z)
• Near-Optimal Bounds for Learning Gaussian Halfspaces with Random Classification Noise [50.64137465792738]
  We show that any efficient SQ algorithm for the problem requires sample complexity at least $Omega(d1/2/(maxp, epsilon)2)$. Our lower bound suggests that this quadratic dependence on $1/epsilon$ is inherent for efficient algorithms.
  arXiv  Detail & Related papers  (2023-07-13T18:59:28Z)
• Fast, Sample-Efficient, Affine-Invariant Private Mean and Covariance Estimation for Subgaussian Distributions [8.40077201352607]
  We present a fast, differentially private algorithm for high-dimensional covariance-aware mean estimation. Our algorithm produces $tildemu$ such that $|mu|_Sigma leq alpha$ as long as $n gtrsim tfrac d alpha2 + tfracd sqrtlog 1/deltaalpha varepsilon+fracdlog 1/deltavarepsilon$.
  arXiv  Detail & Related papers  (2023-01-28T16:57:46Z)
• Sample-optimal classical shadows for pure states [0.0]
  We consider the classical shadows task for pure states in the setting of both joint and independent measurements. In the independent measurement setting, we show that $mathcal O(sqrtBd epsilon-1 + epsilon-2)$ samples suffice.
  arXiv  Detail & Related papers  (2022-11-21T19:24:17Z)
• Learning Distributions over Quantum Measurement Outcomes [4.467248776406005]
  Shadow tomography for quantum states provides a sample efficient approach for predicting the properties of quantum systems. We develop an online shadow tomography procedure that solves this problem with high success probability.
  arXiv  Detail & Related papers  (2022-09-07T09:08:58Z)
• Best Policy Identification in Linear MDPs [70.57916977441262]
  We investigate the problem of best identification in discounted linear Markov+Delta Decision in the fixed confidence setting under a generative model. The lower bound as the solution of an intricate non- optimization program can be used as the starting point to devise such algorithms.
  arXiv  Detail & Related papers  (2022-08-11T04:12:50Z)
• Robust Sparse Mean Estimation via Sum of Squares [42.526664955704746]
  We study the problem of high-dimensional sparse mean estimation in the presence of an $epsilon$-fraction of adversarial outliers. Our algorithms follow the Sum-of-Squares based, to algorithms approach.
  arXiv  Detail & Related papers  (2022-06-07T16:49:54Z)
• Improved Sample Complexity for Incremental Autonomous Exploration in MDPs [132.88757893161699]
  We learn the set of $epsilon$-optimal goal-conditioned policies attaining all states that are incrementally reachable within $L$ steps. DisCo is the first algorithm that can return an $epsilon/c_min$-optimal policy for any cost-sensitive shortest-path problem.
  arXiv  Detail & Related papers  (2020-12-29T14:06:09Z)
• Sample Complexity of Asynchronous Q-Learning: Sharper Analysis and Variance Reduction [63.41789556777387]
  Asynchronous Q-learning aims to learn the optimal action-value function (or Q-function) of a Markov decision process (MDP) We show that the number of samples needed to yield an entrywise $varepsilon$-accurate estimate of the Q-function is at most on the order of $frac1mu_min (1-gamma)5varepsilon2+ fract_mixmu_min (1-gamma)$ up to some logarithmic factor.
  arXiv  Detail & Related papers  (2020-06-04T17:51:00Z)
• Maximizing Determinants under Matroid Constraints [69.25768526213689]
  We study the problem of finding a basis $S$ of $M$ such that $det(sum_i in Sv_i v_i v_itop)$ is maximized. This problem appears in a diverse set of areas such as experimental design, fair allocation of goods, network design, and machine learning.
  arXiv  Detail & Related papers  (2020-04-16T19:16:38Z)

This list is automatically generated from the titles and abstracts of the papers in this site.

This site does not guarantee the quality of this site (including all information) and is not responsible for any consequences.